Cluster Theories and Cluster Structures of Type A

TL;DR

This paper explores various type A cluster theories and structures, introducing a 2D geometric model that generalizes cluster algebras from surfaces and establishing connections among different theories.

Contribution

It introduces a 2D geometric model for type A cluster theories and provides an abstraction that relates multiple existing theories and structures.

Findings

The 2D geometric model generalizes cluster algebras from surfaces.

Several existing type A cluster theories are encoded into this model.

New relations between cluster theories and structures are established.

Abstract

In the present paper we examine the relationship between several type $A$ cluster theories and structures. We define a 2D geometric model of a cluster theory, which generalizes cluster algebras from surfaces, and encode several existing type $A$ cluster theories into a 2D geometric model. We review two other cluster theories of type $A$. Then we introduce an abstraction of cluster structures. We prove two results: the first relates several existing type $A$ cluster theories and the second relates some of these cluster structures using the new abstraction.